CHAPTER 4. MATERIALS AND EXPERIMENTAL PROCEDURE

4.1 Material: Polyamide 12

The materials under study were processed using two different techniques, injection molding (IM) and selective laser sintering (SLS). Specimens made by IM were manufactured by Aries Industrias del Plástico, S.A., located in Spain. The raw material employed was the commercial PA-12 Evonik Vestamid supplied in the form of pellets, and the parameters during the manufacturing process are included in Table 4.1.

The SLS fabrication was carried out by Prodintec, in Spain, employing an EOS Formiga P-100 LS machine which uses a CO₂ laser configured with the optimum process parameters, which are contained in Table 4.2. The commercial PA-12 powder suitable for SLS was neat PA-12 EOS PA2200.

The technical information of the raw materials was supplied by the manufacturers Evonik and EOS. Table 4.3 contains the most remarkable properties of Evonik VESTAMID and EOS PA 2200 reflected in the manufacturers datasheets [135], [136].

Regarding EOS PA 2200, the particle size distribution ranged from 40 to 90 μm, being centred on 50 – 60 μm.

Different specimen configurations were manufactured. In the case of SLS, due to the anisotropic nature of the process, two batches of specimens were fabricated for each configuration. In the one named 0°, the layered structure was disposed in such a way that the force was applied parallel to the layer planes; meanwhile in the batch named 90°, the layers were perpendicular to the applied force (Figure 4-1).

4.2 Physical characterization

4.2.1 Density measurement

Despite the theoretical densities provided by the manufacturers, the experimental ones were measured in the specimens produced. The density of the pieces was determined experimentally measuring 5 specimens of each geometry in a precision balance Mettler Toledo AX205 DeltaRange^®, which has a resolution of ± 0.00001 g. Following the Archimedes principle, each specimen was introduced in a beaker with a known immersion medium and measuring the amount of displace fluid. The density of the sample, ρ, is given by the following expression:

$$\rho \mathit{=}\frac{{\mathit{W}}_{\mathit{\text{air}}}}{{\mathit{W}}_{\mathit{\text{air}}}\mathit{-}{\mathit{W}}_{\mathit{\text{im}}}}\mathit{(}{\mathit{\rho }}_{\mathit{0}}\mathit{-}{\mathit{\rho }}_{\mathit{L}}\mathit{)}\mathit{+}{\rho }_{\mathit{L}} (4-1)$$

where W_air is the weight of sample in air, W_im is the weight of sample in the immersion medium, ρ₀ is the density of the immersion medium and ρ_L the density of air (0.0012 g/cm₃). Due to the nearness of the expected PA-12 density to that of distilled water, commonly used as immersion medium, which is ρ₀ = 0.99782 g/cm³, acetone was also employed as immersion medium in this case, with p₀ = 0.787 g/cm³. The closed porosity of each batch was calculated in reference to the theoretical bulk density value of 1.02 g/cm³ of the PA-12 [137]. Five replicas were measured for each material and condition.

4.2.2 Surface Roughness tests

Surface roughness was measured using a roughness tester Mitutoyo SJ-301 equipped with a detector whose tip had a radius of 60 μm and a tip angle of 60°. A total number of 62 tests were carried out in the dumbbell geometry of each material longitudinally and transversely. The profile roughness parameters obtained were the maximum peak to valley height of the profile, R_z, and the arithmetical mean value of the tested profile along a length l_r, R_a, given by:

$$R_{\mathit{a}}\mathit{=}\frac{\mathit{1}}{{\mathit{I}}_{\mathit{r}}}{\mathit{\int }}_{\mathit{0}}^{\mathit{l}\mathit{r}}\mathit{ }\left|z\mathit{\text{(}}x\text{)}\right|dx (4-2)$$

4.3 Microstructural characterisation

The microstructural characterisation aims to analyse the crystalline phase, focusing on its morphology and its size. For SLS PA-12, films between 3 and 15 μm in thickness were sectioned from the centre of the bulk using a rotary microtome Leica RM2255 [138]–[140]. The extractions were performed parallel and perpendicular to the build direction. Although the anisotropy is not a key microstructural factor in IM specimens, an initial optical inspection revealed some differences between the core and the skin of the material. The same procedure as in SLS material was followed for IM, that is, the films were extracted from either the core or the skin of the bulk specimen. These films were placed on microscope glass slides and analysed in an optical microscope Motic BA310 Met-T equipped with a 5 megapixels ZEISS Axiocam 105 camera. For their inspection, transmitted polarized light and dark field modes were applied [141]. A total number of 90 micrographs were explored, 30 of SLS at 0° orientation, 40 of SLS 90° at orientation and 20 of IM. The images of the SLS specimens were taken using a 50x objective with a 10x eyepiece lens, whereas for the examination of the IM films, a x100 objective lens immersed in a transparent oil with a 1.482 refraction index to attain a total magnification of x1482 was needed for the visualization of the characteristic microstructural features.

A total number of 157 spherulites were measured manually using an image analysis software.

4.4 Thermal characterisation

The purpose of the thermal characterisation was to determine the thermal properties and the crystallisation characteristics of SLS PA-12 and IM PA-12.

Firstly, a Differential Scanning Calorimetry (DSC) was performed with a DSC Mettler 822e equipment. DSC tests consisted in using specimens with 12.0 ± 0.5 mg in weight which were subjected to a heating ramp at a rate of 10 °C/min from 25 °C to 250 °C, followed by a cooling ramp at a rate of 10 °C/min from 250 °C to 25 °C. This heating/cooling processes were performed twice in a row. The crystallisation temperature, T_c, and the enthalpy, ΔH_c, were collected from the cooling cycle, meanwhile the melting temperature T_m and the enthalpy, ΔH_m, and, when detectable, the glass transition temperature, T_g, were extracted from the second heating cycle.

Regarding the calculation of the crystallinity degrees, it was distinguished between the melting crystallinity degree, χ_m, and the crystallinity degree from the crystallisation peak, χ_c. In case of the former, the heat of fusion, △H_m, was obtained by integrating the heat flow under the melting peak from the second heating cycle; and for the latter and in a similar way, the crystallization heat, △H_c, was also calculated by integrating under the crystallisation peak from the cooling cycle. Finally, the crystallinity degrees were computed using the following equations:

$${\chi }_{\mathit{m}}\mathit{=}\frac{\mathit{\Delta }{\mathit{H}}_{\mathit{m}}}{\mathit{\Delta }{\mathit{H}}_{\mathit{0}}}\mathit{;}\chi c\mathit{=}\frac{\mathit{\Delta }{\mathit{H}}_{\mathit{c}}}{\mathit{\Delta }{\mathit{H}}_{\mathit{0}}} (4-3)$$

using an enthalpy of fusion, △Ho = 209 J/g, which is the enthalpy of fusion of perfect PA-12 crystals [142]. The crystallinity degrees were calculated as the average value from χ_m and χ_c.

Additionally, Dynamic Mechanical Analysis (DMA) was carried out using a TA Instruments DMA Q800 analyser with a single cantilever configuration according to ASTM D5023 Standard [143]. Specimens with 17×12×3 mm³ in size were heated from - 100 °C to 150 °C at 1 Hz of frequency and at a heating rate of 3 °C/min. The glass transition temperatures were computed as the maximum values of the loss factor (tanδ) curves.

4.5 Mechanical characterisation

Tensile tests were carried out according to ASTM D638 Standard [144] to determine mechanical properties such as tensile strength, Young’s modulus, Poisson’s ratio and elongation at break. These tests were done using dumbell type IV specimens with the dimensions shown in Figure 4-2.

Tests were carried out at -50 °C, 23 °C and 50 °C at 0° and 90° orientations in SLS specimens and at 23 °C in IM samples. A universal electromechanical testing machine MTS Alliance RF/100 equipped with a load cell of ± 5 kN was employed. The crosshead speed was 5 mm/min and a contact extensometer MTS 634.12F-54 (Figure 4-3) was used for measuring the axial deformation. A total number of three valid repetitions were performed for each condition. For the tests at -50 °C and 50 °C, the load train, formed by the hinges, grips and the specimen with the extensometer, were placed inside an environmental chamber MTS 651.06E-03 (Figure 4-3). Cooling and heating processes were performed maintaining a constant load of 100 N on the specimen for balancing the thermal contractions. When the required temperature was reached, conditioning was held for a minimum of 30 minutes to reach a stationary state before starting the test.

In addition, 2-Dimensional video correlation was used at all temperatures for obtaining the 2D displacement field and, in particular, the longitudinal and the transverse deformation using a VIC 2D videoextensometer. This equipment consisted of a PointGrey Grasshopper3 5 Megapixel camera connected to a computer with the software Correlated Solutions VIC 2D. Because videoextensometry needs reference points to follow their movement during the test, the surface of the white specimens were sprayed with black paint to attain a random, matte pattern of speckles, providing a large quantity of points to be analysed, as shown in Figure 4-4.

Moreover, to obtain the best possible contrast in the images, LED lightening systems were employed to avoid any type of reflection of the specimen surface. The camera must be focused on the specimen perpendicularly to its lateral surface. The tensile test assembly at room temperature is shown in Figure 4-5.

4.6. Fracture and fatigue tests

4.6.1 Crack sharpening

Compact Tension (CT) configuration with the dimensions shown in Figure 4-6 was used for fracture and fatigue tests. The only difference between fracture and fatigue specimens was the initial notch length, which in the fatigue specimens was the half that in fracture ones in order to fulfil ASTM D5045 standard [145] in fracture and ASTM E647 standard [146] in fatigue characterisations.

The sharp crack in the fracture CT specimens was generated by tapping a razor blade with a thickness of 0.3 mm, previously frozen at liquid nitrogen temperature (-196 °C), on the root of the machined notch till attaining an initial natural crack length to width ratio, a₀/W, of 0.5. This ratio ensures a sharp crack length enough to avoid the influence of the manufactured notch on the results [43], [145].

In the fatigue CT specimens, the sharp crack was introduced by tapping in a similar way as in the fracture samples but, in this case the sharp crack lengths should be only larger than the notch height, of 2 mm, to avoid the notch effects [43], [146].

Small cracks were introduced in dumbbell specimens with lengths of 0.2 mm, 0.3 mm, 0.5 mm, 1 mm, 2 mm, 2.5 mm and 3 mm by pressing a microtome razor blade with a tip diameter of 5.3 μm (Figure 4-7). This tip diameter ensures a notch radius smaller than 10 μm, which is the estimated limit for considering the crack as a natural crack [147], [148]. These specimens were used for static and fatigue characterisations in the PSC regime (section 2.4)

After notch sharpening or small crack introduction, the crack front was inspected either visually or by optical means to guarantee no damage in form of microcracks or whitening. In case of presence of damage, the samples were discarded.

4.6.2 Fracture tests

Tests were carried out in a universal servohydraulic testing machine MTS 810 Materials Testing, equipped with a load cell of ± 5 kN. The crosshead speed was 5 mm/min. A Crack Opening Displacement transductor (COD) MTS 632.02F-20 with a displacement range of + 3.9 mm/ -2 mm was also employed. The tests were performed in SLS specimens at 0° and 90° orientations, and in IM samples. To study the evolution of the fracture properties with the temperature, tests were undertaken at three different temperatures, -50 °C, 23 °C and 50 °C, completing three repetitions of each condition. For the -50 °C and 50 °C tests, an environmental chamber MTS 651.06E-03 was used, installing the load train, formed by the hinges, grips and the specimen with the COD transductor, inside it (Figure 4-8).

Similar procedure to that of the tensile tests was performed, maintaining a constant load of 50 N on the specimen during cooling and heating processes until the load frame finished of balancing the thermal contractions and, consequently, reaching a stationary state. When the required temperature was reached, conditioning was held 25 min more before starting the test to guarantee the thermal equilibrium.

Attending to the mechanical behaviour of each material and test condition, the results were analysed applying the Linear Elastic Fracture Mechanics (LEFM) or the Non-Linear Fracture Mechanics (NLFM) approach as appropriate. The ISO 13586 Standard was followed to determine the crack opening mode (mode I) fracture toughness in terms of the stress intensity factor, K_IC, and the energy release rate, G_IC, under LEFM approach [149].

From the load (P)-crack mouth opening displacement (v) record, the maximum load, P_max, and a conditional load, P_Q, are calculated. P_Q is used to determine a conditional stress intensity factor, K_Q, and a conditional energy release rate, G_Q.

Firstly, P_5% is computed as the intersection of the P-v record with a line having a 5% smaller stiffness than the initial one, S. If the maximum of the curve falls within the best straight line to determine S and the line with a 5% lower stiffness, then Pmax corresponds with the load at crack growth initiation, P_Q. If P_max falls outside these two lines, then P_Q is P_5%. If P_max/P_Q < 1.1 then P_Q is used for K_Q calculation, otherwise, the test is invalid. With this limitation, the critical stress intensity factor K_Q is obtained:

$$K_{\mathit{Q}}\mathit{=}f\mathit{\left(}\alpha \mathit{\right)}\frac{{\mathit{P}}_{\mathit{Q}}}{\mathit{B}\sqrt{\mathit{W}}} (4-4)$$

with B the specimen thickness, W the width and f(α) the geometry calibration factor depending on the crack length a given by:

$$\begin{array}{c}\mathit{f}\mathit{\left(}\mathit{\alpha }\mathit{\right)}\mathit{=}\frac{\mathit{(}\mathit{\text{2}}\mathit{+}\mathit{\alpha }\mathit{)}}{\mathit{(}\mathit{\text{1}}\mathit{-}\mathit{\alpha }{\mathit{)}}^{\mathit{3}\mathit{/}\mathit{2}}}(\mathrm{0}\mathrm{.}\mathrm{886}\mathrm{+}\mathrm{4}\mathrm{.}\mathrm{64}\alpha -\mathrm{13}\mathrm{.}\mathrm{32}{\alpha }^{\mathrm{2}}\\ +14.72{\alpha }^3-5.6{\alpha }^4)\end{array} (4-5)$$

being α=a/W.

In addition to the previous condition, the standard establishes that to guarantee the plane-strain state, it must be fulfilled:

$$B,W-a,a\ge 2.5\frac{K_Q^2}{{\sigma }_y^2} (4-4)$$

with σ_y the yield stress of the material for the temperature and loading rate of the test and K_Q is K_IC in this case. The energy release rate, G_Q is G_IC and in principle it can be obtained from the following equation:

$$G_{IC}=\frac{\left(1-v^2\right){K_{IC}}^2}{E} (4-7)$$

E is the Young’s modulus and v the Poisson’s ratio obtained at the same time and temperature conditions. Due to the many uncertainties introduced by this procedure, particularly, in the determination of the Young’s modulus, G_Q is computed from:

$$G_Q=\frac{U_Q}{BW\phi \left(\alpha \right)} (4-8)$$

where U_Q is the energy obtained from the area under the P-v curve until the point P_Q and ϕ (a/W) is the energy calibration factor depending on the crack length given by this expression:

$$\phi (a/W)=\frac{A^{\prime }(1-\alpha )}{H+2A^{\prime }} (4-9)$$

with

$$\begin{array}{c}{A}^{\prime }=1.9118+19.118\alpha -2.5122{\alpha }^2-23.226{\alpha }^3\\ -20.54{\alpha }^4\end{array} (4-10)$$

and

$$\begin{array}{r}H=(19.118-5.0244\alpha -69.678{\alpha }^2\\ \left.+82.16{\alpha }^3\right)(1-\alpha )\end{array} (4-11)$$

When the behaviour of the material deviates from LEFM, the J-integral vs. crack growth resistance (J-R) curves were obtained directly from the force displacement record according to ASTM E1820 standard using the normalization method [150]. The starting point of this procedure is the measurement, by optical means, of the initial crack length, a_o, and the final crack length, a_f, taken from the fracture surface. Then, each force value P_i is normalized up to but without including the maximum value P_max, following the expression:

$$P_{Ni}=\frac{P_i}{WB{\left[\frac{W-a_{bi}}{W}\right]}^{\eta pl}} (4-12)$$

where Ƞ_pl is the plastic constriction factor, which depends on the specimen geometry and is given by:

$${\eta }_{pl}=2+\frac{0.522\left(W-a_0\right)}{W} (4-13)$$

and a_bi is the crack growth applying the blunting correction:

$$a_{bi}=a_0+\frac{J_i}{2{\sigma }_y} (4-14)$$

where

$$J_i=\frac{K_i^2\left(1-v^2\right)}{E}+J_{pl,i} (4-15)$$

with K_i the stress intensity factor obtained from eq. (4-4) and eq. (4-5) and J_pl,i the plastic component of the J-integral:

$$J_{pl,i}=\frac{{\eta }_{pl}{U}_i}{B\left(W-a_0\right)} (4-16)$$

where U_i is the area under the curve until P_i.

The displacement corresponding with each force level P_i, δ_i, is also normalized to use a plastic normalized displacement, δ’_pli:

$${\delta }_{pli}^{\prime }=\frac{{\delta }_{pli}}{W}=\frac{{\delta }_i-P_i{C}_i}{W} (4-17)$$

being C_i the specimen elastic load-line compliance using the crack length a_bi. The same procedure is employed to obtain the last point, at P_max, but using, the measured final crack length, a_f, instead of a_o, the P_Ni-δ’_pli values are fitted to the normalized function:

$$P_N=\frac{a+b{\delta }_{pl}^{\prime }+c{\delta }_{pl}^{\prime 2}}{d+{\delta }_{pl}^{\prime }} (4-18)$$

with a, b, c and d the fitting parameters. Once these parameters are determined, a_i values are calculated using an iterative method.

Finally, with the force, the displacement and the crack size estimated at each point, the J integral value for each i point was determined using:

$$J_i=\frac{\eta U_i}{B\left(W-a_i\right)} (4-19)$$

With Ƞ = Ƞ_pl for the CT configuration and the crack extension, Δa, as:

$$\mathrm{\Delta }a=a_i-a_0 (4-20)$$

The resulting J-crack growth resistance curve is fitted to a power law J = C_jΔa^N_J with N_J < 1. The crack initiation resistance, J_IC, is calculated as the minimum value between J_0.2, which defines crack resistance at 0.2 mm of the total crack growth, and J_BL, computed as the intersection between the crack growth resistance curve and the blunting line defined as:

$$J_{BL}=2{\sigma }_y\mathrm{\Delta }a (4-21)$$

The size requirements for plane strain J_IC is given by [151]:

$$B,a,W-a>25\frac{J_{1C}}{{\sigma }_y} (4-22)$$

The normalized method was applied using the MATLAB^® script included in the Annex A.1.

4.6.3 Fatigue crack growth tests

The fatigue tests were carried out under the recommendations of the ASTM E647 Standard [146] and ESIS TC4 protocol [152], which aim to obtain the crack propagation threshold value of the control parameter and the fatigue crack propagation curves.

A total number of eight samples of IM material and fourteen samples of each SLS orientation were tested at room temperature in a servo-hydraulic testing machine MTS 810 Materials Testing with a load cell of ± 5 kN and a crack opening displacement (COD) extensometer MTS 632.02 F-20 with a displacement range of +3.9 mm/-2 mm (Figure 4-9). The tests were carried out at room temperature (23°C), with a frequency of 1 Hz, load ratio R = P_min/P_max of 0.1 and imposing a sinusoidal load wave. Two test procedures were used according to ASTM E647.

To obtain the crack propagation behaviour until failure, five IM specimens and ten SLS specimens per orientation were tested in constant-force-amplitude configuration or increasing control parameter mode. On the other hand, to obtain the crack propagation threshold, three IM specimens and four SLS specimens per orientation were tested under the decreasing control parameter procedure. Beginning with a control parameter value under the critical value obtained from the fracture tests, tests with decreasing load amplitude were carried out with a step of decrease or normalized variation of the control parameter of -0.05 mm^-1. The threshold values were computed for crack growth rate downs to 3 • 10^-7 mm per cycle.

Two methods were used for obtaining the crack growth during the tests. In the first method, the crack growth was calculated by the compliance method from the flexibility of the specimen using expressions determined for metals but also validated for polymers [152]. The expressions that relate the crack length to width ratio, a/W, with the compliance in CT specimens are

$$\begin{array}{c}\frac{a}{W}=1.001-4.6695U+18.460U^2+236.82U^3\\ +1214.9U^4-2143.6U^5\end{array} (4-23)$$

where:

$$U=\frac{1}{{\left(\frac{BEv}{P}\right)}^{1/2}+1} (4-24)$$

To validate the compliance method, an optical method was used. Images were captured during the test at a specific frequency allowing the tracking of the crack length with a millimetre scale stuck to the specimen’s surface. Moreover, the calculated final crack length was compared with the measured fatigue crack length on the fracture surface of each specimen.

The fatigue behaviour has been determined as the crack growth rate, da/dN, versus the Crack Driving Force, CDF, $\mathrm{\Delta }\sqrt{G}$ according to equation (2-2).

The crack growth rate at any average crack length $\overline{\mathrm{a}}=\frac{a_{i+1}-a_i}{2}$ is obtained from the crack length versus elapsed cycle with the following expression:

$${\left(\frac{da}{dN}\right)}_{\overline{a}}=\frac{a_{i+1}-a_i}{N_{i+1}-N_i} (4-25)$$

The control parameter $∆\sqrt{G}$ was obtained from equation (2-1) in which the value of dC/da is given by:

$$\frac{dC}{da}=\frac{1}{EW}\frac{dg}{d\alpha } (4-26)$$

where g(α) is the non-dimensional load line compliance [26] and with dg/dα for CT specimens computed as:

$$\begin{array}{c}\frac{dg}{d\alpha }=\frac{(1+\alpha )}{(1-\alpha {)}^2}x\\ \left[\frac{4}{(1-\alpha )}\left(2.163+12.219\alpha -20.065{\alpha }^2-0.9925{\alpha }^3-20.609{\alpha }^4-9.9314{\alpha }^5\right)\right.\\ \left.+(1+\alpha )\left(12.219-40.13\alpha -2.9775{\alpha }^2+82.436{\alpha }^3-49.657{\alpha }^4\right)\right]\end{array} (4-27)$$

4.6.4 Fatigue life tests

Fatigue life tests were performed to obtain S-N curves for IM and SLS specimens at both orientations, according to ASTM D7791 Standard [153]. The specimens had the same geometry and dimensions as those utilized in tensile tests (Figure 4-2). A total number of 89 specimens were tested, of which 20 were used for the determination of IM PA-12, 32 for SLS PA-12 at 90° orientation and 37 for PA-12 at 0° orientation. Tests were performed at room temperature in a servo-hydraulic testing machine MTS 810 Materials Testing with a load cell of ± 5 kN. A contact extensometer MTS 634.31F-24 with a displacement range of +4 mm/-2 mm was employed to measure the axial strain.

The frequency was 1 Hz, using a sinusoidal load wave form at a constant load amplitude with a load ratio, R, of 0.1. Tests were done at 12 different stress levels, that is with the maximum stress of the cycle, σ_max, to the tensile strength ratio between 58.75% and 85%, to guarantee to be within the High Cycle Fatigue regime. At least, two repetitions for each stress level were carried out.

The tests at the lowest levels were finished at 10⁶ cycles due to their high timeconsuming aspect, establishing tests which reached this number of cycles without failure as run-outs. Each test was plotted as a single point of the stress range, Δσ, versus N representation, giving as result the Wöhler curve for each material.

The fatigue life curves were fitted to the Basquin equation:

$$\mathrm{\Delta }\sigma =B_B{N}^{mB} (4-28)$$

The estimated value of the fatigue limit, Δσ_fl, at 10⁶ cycles were determined by the maximum likelihood method, following a test sequence based on the staircase method. In this method, fatigue tests were sequenced at different stress levels with a fixed difference in the applied stress range or initial step of 2.5% of the tensile strength. If a specimen failed before reaching the end of the test, the following test was carried out at a stress level one step lower. Otherwise, the stress level was increased one step in the following test. To obtain a more accurate value, the step was decreased after seven completed tests to 1.25% of the tensile strength. Although the change in the step removed the possibility of using the Staircase method equations developed for metals [154], the maximum likelihood method allowed to obtain an accurate estimation of Δσ_fl [155]–[157]. A total number of 17 for SLS PA-12 specimens at 0° orientation, 12 for PA-12 samples at the 90° orientation and 14 for IM material were tested to obtain Δσ_fl.

The maximum likelihood method was applied assuming that the stress levels of the run-out specimens follow a normal distribution [156], [158]. For a normal distribution with a mean value, μ, and a standard deviation value, s, the likelihood function, L, is given by:

$$L=\prod _{i=1}^{n_T} {\left[1-F\left(\mathrm{\Delta }{\sigma }_i,\mu ,s\right)\right]}^{r_i}{\left[F\left(\mathrm{\Delta }{\sigma }_i,\mu ,s\right)\right]}^{f_i} (4-29)$$

where r_i are the number of run-outs and f_i are the number of failures at the i^th stress level, Δσ_i, n_T is the total number of stress amplitude levels and F is the cumulative probability of failure of the normal distribution.

This function depends on the stress at the i^th stress level, Aσ_i and the parameters of the normal distribution and is calculated with the following expression:

$$F\left(\mathrm{\Delta }{\sigma }_i,\mu ,\sigma \right)={\int }_{-\mathrm{\infty }}^{\mathrm{\Delta }{\sigma }_i} \frac{1}{s\sqrt{2\pi }}\exp \left[-\frac{1}{2}{\left(\frac{\mu -x}{s}\right)}^2\right]dx (4-30)$$

The maximum of the likelihood function was obtained using the MATLAB^® script proposed by Meizoso et al. [155] and included in the Annex A.2.

4.6.4.1 Determination of fracture parameters from damage evolution in plain fatigue samples using the Fracture Mechanics methodology

The fracture toughness was calculated from the measurement of the damage observed and measured from the analysis of the morphology of the fracture surfaces resulting of the fatigue life tests in SLS PA-12. The extent of the damage was delimited, that is, the limit associated with the area of crack propagation before instability occurred was determined from the optical and Scanning Electron Microscopy (SEM) inspections of the fracture surfaces. The crack nucleated at some point at the surface due to the elevated roughness resulting from the manufacturing process.

The geometrical configuration of the fatigue specimens with a crack growth zone resulting from the crack propagation stage before failure could be assimilated to the Single Edge Notch Tension (SENT) fracture specimens.

The fracture parameters were determined using the NLFM approach and the J-integral at the instant of failure was computed as

$$J=\frac{\eta U}{B(W-a)} (4-31)$$

where U is the area under the unloading load-displacement record of the last cycle before failure, a is the length of the subcritical crack growth zone measured directly from the fracture surface and Ƞ is a plastic geometric factor that in case of the SENT configuration is defined as [159]:

$$\begin{array}{r}\eta =1.067-1.767\left(\frac{a}{W}\right)+7.808{\left(\frac{a}{W}\right)}^2\\ -18.269{\left(\frac{a}{W}\right)}^3+15.295{\left(\frac{a}{W}\right)}^4\\ -3.083{\left(\frac{a}{W}\right)}^5\end{array} (4-32)$$

4.7 Fractography

The fracture surfaces of the tensile, fracture, fatigue crack growth and fatigue life specimens were inspected using scanning electron microscopy. The aim was to determine the micromechanisms of failure dominant at different temperatures, orientations and different manufacturing conditions. The fracture surfaces were gold coated to enhance their conductivity using a sputter coater EMITECH K550X with a current of 30 mA for 1 minute. The coated samples were positioned on a conductive specimen holder and examined in a SEM HITACHI S-3400N.

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